How can I join the geometry of intersecting faces/planes and then remove parts?
Separate, add Boolean Modifiers, then Rejoin

You can achieve this by:


Separating the different planes into different objects. 
Adding a Boolean modifier set to either Union or Intersect on both of your planes, then applying both modifiers. The result is that both planes will have perfectly made edges where they intersect. 
You can then Join them into one mesh and edit parts of each as you wish without disrupting the textures of either plane.


For example, here there are 2 intersecting planes, both part of one object:



Here's how you begin:


Select the object, enter Edit mode (TAB) and then separate each plane to be it's own object with the following method:  


Select one plane → Separate (P) → Selection 


Repeat this until each plane has been separated, then exit Edit mode with TAB.
NOTE: Do not move either plane from it's location so they will still be in the proper place when you rejoin them back to the larger object at the end. 
Add a Boolean modifier to each plane:
Select plane → Object Properties panel → Modifiers tab → Boolean 


Set the Operation to Union or Intersect
Set the Object to the other plane


For example, here is the modifier added to the "Blender Logo plane":


And here is the modifier on the "Blank plane":

Click the Apply button on both modifiers
Select both planes and the larger structure they were part of, then use Join (CTRLJ) to join them back into one mesh again 

The result is you now have one mesh object with perfectly marked edges where the planes intersected, and you can edit the faces as you wish without disrupting the textures:




NOTE: I'm using 2.70a and there may be a bug in the Boolean modifiers: I noticed that sometimes when I clicked "Apply" on the modifiers, the planes disappeared sometimes, and other times they behaved oddly. If this happens, just use Undo (CTRLZ) to go back to before you clicked "Apply" and try again. Eventually it worked for me.
:)